Derivatives¶
Motivation¶
- Problem: Measuring velocity of car at given time
- Solution: Estimate average velocity given 2 points (secant), average = rise / run
- Secant: A line that intersects a curve at a minimum of two distinct points


Derivatives and Tangents¶
- Solution to above problem: Move the distance between rise and run so close that you can't tell the distance between the two.
- Derivate: Slope of the tangent line at a specific point in a function, $dx/dt$
- Instantaneous rate of change: Slope of the tangent line

Slopes, maxima and minima¶
Zero Slope¶

Maxima / Minima¶
Takeyaways:
- You can find the maxima / minima of a function by finding the zero slope (ie. tangent line is horizontal)

Derivatives and Notation¶
Lagrange vs Leibniz Notation

Some Common Derivatives¶
Constant (horizontal line)¶

Line (diagonal straight line)¶
Solution: $\frac{\Delta y}{\Delta x}$

Quadratics¶
Deriving the slope at (1, 1):

Formula:

Higher-degree polynomials¶
Cubed¶
Deriving the slope at (0.5, 0.2):

Formula:

Other Power Functions¶
Calculating the slope between (1,1) and (1.5, 2/3):

Deriving the slope at the limit of (1,1):

Formula:

Summary¶
Pattern for power functions:
- Exponent becomes the derivative as a multiplication factor
- Subtract one from the exponent and you get the new exponent

Common Derivatives¶

Inverse Function¶
What's an inverse?¶
- Undoes the original function $f(x)$ by applying $g(x)$.
- For $x$ becoming $x^2$, $g(x)$ applies a square root to $x^2$
- Notation on the right (notice the $-1$ on $f$

Derivative of the Inverse¶
Takeaways:
- $f(x)$ and $g(x)$ mirror each other
- Their slopes, $\frac{\Delta f}{\Delta x}$ and $\frac{\Delta g}{\Delta y}$ essentially mirror each other
- Therefore, the derivative of $g$ ($g'(y)$) is the inverse of the derivative of $f$ ($f'(x)$):
$$
g'(y) = \frac{1}{f'(x)}
$$

Using an example at (1,1):

Using an example (2, 4):

Derivative of Trigonometric Functions¶
Sine¶

Cosine¶

Meaning of the Exponential $(e)$¶
$$
e = 2.71828182
$$
Convergence of $e$:
- Another way to define $e$ is $(1 + \frac{1}{n})^n$
- As $n$ reaches infinity, it becomes $e$
- Derivative of $e$ is the same ($e^x$)

Intuition using bank interest as an example:

Derivative of $e^x$ is $e^x$¶
Proving derivative using secants:

Derivative of $log(x)$¶
Properties of logarithm¶
- Logarithm of $x$ ($log(x)$) is the inverse function of $e^x$
- The derivative of $f(x) = e^x$ as (2, 7.89) is $e^2$
- The derivative of its inverse (7.89, 2), ie $log(y)$ is $\frac{1}{e^2}$
- The derivative of $log(y)$ is $\frac{1}{y}$

Existence of the Derivative¶
Non-differentiable functions: Functions where you can't calculate a derivative at every point
- For a function to be differentiable, the derivative has to exist for every point in the interval
- Non-differentiable functions have a point where any tangent could work.
- Examples: absolute ($|x|$), piecewise functions, functions with a vertical tangent ($f(x)=x^\frac{1}{3}$)
